R E S E A R CH AR T I C L E

Regulatory feedback on receptor and non-receptorsynthesis for robust signaling

Arthur D. Lander1,2 | Qing Nie1,2,3,4 | Cynthia Sanchez-Tapia5 |

Aghavni Simonyan3,6 | Frederic Y. M. Wan3

1Department of Developmental and CellBiology, University of California Irvine,Irvine, California2Center for Complex Biological Systems(CCBS), University of California Irvine,Irvine, California3Department of Mathematics, Universityof California Irvine, Irvine, California4NSF-Simon Center for Multiscale CellFate (CMCF), University of CaliforniaIrvine, Irvine, California5Department of Mathematics, RutgersUniversity, Piscataway, New Jersey6Geffen Academy, UCLA, Los Angeles,California

CorrespondenceFrederic Y. M. Wan, Department ofMathematics, UC Irvine, Irvine, CA92697-3875.Email: fwan@uci.edu

Funding informationAir Force Office of Scientific Research,Grant/Award Number: FA9550-14-1-0060;National Institues of Health, Grant/AwardNumber: P50-GM-076516; NationalScience Foundation, Grant/AwardNumbers: DMS-1129008, DMS176-3272;Simons Foundation, Grant/AwardNumber: DMS-176-3272

Abstract

Elaborate regulatory feedback processes are thought to make biological devel-

opment robust, that is, resistant to changes induced by genetic or environmen-

tal perturbations. How this might be done is still not completely understood.

Previous numerical simulations on reaction-diffusion models of Dpp gradients

in Drosophila wing imaginal disc have showed that feedback (of the Hill func-

tion type) on (signaling) receptors and/or non-(signaling) receptors are of lim-

ited effectiveness in promoting robustness. Spatial nonuniformity of the

feedback processes has also been shown theoretically to lead to serious shape

distortion and a principal cause for ineffectiveness. Through mathematical

modeling and analysis, the present article shows that spatially uniform non-

local feedback mechanisms typically modify gradient shape through a shape

parameter (that does not change with location). This in turn enables us to

uncover new multi-feedback instrument for effective promotion of robust sig-

naling gradients.

1 | INTRODUCTION

At some (early) stage of embryonic development of a bio-logical organism, one or more proteins (known as mor-phogens or ligands) responsible for cell differentiation aresynthesized and transported away from their sources to bebound to relevant cell receptors at different locations toform signaling morphogen-receptor complexes, known asa signaling (spatial) gradients. Such signaling gradients

convey positional information for cells to adopt differentialfates to result in tissue patterns. This process of cell differ-entiation is well established in developmental biology. Forexample, the morphogen Decapentaplegic (Dpp) involvedin the development of the Drosophila wing imaginal discis synthesized in a narrow region at the boundary betweenthe anterior and posterior compartments of the disc. Dppmolecules produced are transported away from the local-ized source and flow out of the disc upon reaching its

Received: 2 December 2019 Accepted: 16 January 2020

DOI: 10.1002/dvdy.160

Developmental Dynamics. 2020;249:383–409. wileyonlinelibrary.com/journal/dvdy © 2020 Wiley Periodicals, Inc. 383

edge. Some Dpp molecules bind reversibly with the cell-surface signaling receptor Thickvein (Tkv) to form a spatialgradient of signaling morphogen concentration over thespan of the wing imaginal disc. Graded differences inreceptor occupancy at different locations underlie the sig-naling differences that ultimately lead cells down differentpaths of development.1-5 Simple models of this process ofgradient formation have been shown theoretically to pro-duce a unique signaling gradient that is monotone andasymptotically stable with respect to small (one time) per-turbations (see References 6 and 7 for example).

For normal biological development, it is importantthat signaling morphogen gradients not be easily alteredby (sustained) genetic or epigenetic effects on the consti-tution of the biological organism.8 Experiments (carriedout by S. Zhou in A.D. Lander's lab (see also Reference9)) have shown that Dpp synthesis rate doubles when theambient temperature is increased by 5.9 !C. With suchan increase in Dpp synthesis rate, the simple modelsdeveloped in References 6, 7, and 10 would predict anenhanced or (more commonly called) “aberrant” (or“abnormal”) signaling gradient quantitatively and quali-tatively different from that under the (lower) normalambient temperature. Yet development of the wing imag-inal disc generally does not change significantly withtemperature changes of such magnitude. The insensitiv-ity of system output to sustained alterations in input orsystem characteristics so necessary for normal develop-ment is often termed robustness of biological develop-ment. How this robustness requirement is met has beenthe subject of a number of recent studies.11-22

Evidence exists that regulatory feedback processes playa role in rendering biological developments robust, that is,a signaling gradient resistant to changes expected to beinduced by genetic or environmental perturbations.23,24

How this might be accomplished is still not completelyunderstood. Among the first attempt to determine mecha-nisms for attaining robust developments, a negative feed-back on receptor synthesis rate was investigated inReference 25. A Hill function type negative feedback wasincorporated into the basic morphogen gradient model ofReference 7 to reduce the synthesis rate of Tkv by anamount that depends on the aberrant signaling morphogenconcentration. Some numerical simulations of the modelshow that robustness of the signaling gradient (and hencethe corresponding development) with respect to a sustainedgenetic or epigenetic perturbation is not achieved for any ofthe 106 combinations of system parameter values in aparameter space of 6-dimensions. A subsequent theoreticalanalysis delineated and confirmed the ineffectiveness of thisnegative feedback mechanism.26 Briefly, a Hill type negativefeedback reduces the receptor synthesis rate nonuniformly,

disproportionately more so at locations of high signalingmorphogen concentration. Such reduction generally leads toa modified gradient of different slope and convexity fromthe normal (wild-type) gradient. The theoretical results sug-gest that a spatially uniform negative feedback respondingto some overall measure of abnormality or aberrancy (suchas the average impact of the local changes on the system)may be more effective.26 This suggestion has led to the initi-ation of a new general approach to attain robust develop-ment by way of feedback mechanisms that are spatiallyuniform.20-22,27

With a view that most feedback mechanisms have theultimate effect of reducing the morphogen available forbinding with signaling receptors, a proof-of-concept proto-type model for a spatially uniform negative feedback onmorphogen synthesis rate was first investigated in Reference27. The findings in that preliminary effort provided theimpetus to investigate in Reference 22 the efficacy of spatialuniformity in other known feedback mechanisms. In thisarticle, we refine the models for modifying an aberrant sig-naling morphogen gradient (toward the original wild-typegradient) investigated in References 22 and 27 by modelingexplicitly one of the processes known to reduce the availabil-ity of unbound morphogen molecules. Among the differentways to reduce Dpp concentration is their binding withother protein molecules to form morphogen complexesthat do not signal for cell differentiation.28 Such (non-sig-naling) companion proteins are known to exist for Dppand other BMP family ligands. They include Notum,29

Nog (noggin),30-32 Chd (chordin),33,34 Dally (division abnor-mally delayed),35,36 FST (follistatin),37-40 Sog (shortgastrulation),41,42 and various heparan sulfate proteogly-cans.43 Collectively, they are called non-receptors since theybind with morphogens such as Dpp but the resulting boundmorphogen complexes have no role in cell differentiation.

Effects of non-receptors was modeled and analyzed inReference 44 where we extend the simple wing disc mor-phogen model of References 6 and 7 to include a fixedconcentration of cell-surface non-receptor (inducedinstantaneously at the onset of the genetic or epigeneticperturbations at time te). This simplest model offered thefirst theoretical glimpse into the inhibiting effects of non-receptors on the formation and properties of steady statesignaling morphogen-receptor gradients. Subsequently,large scale computational studies of non-receptors syn-thesized at a prescribed (perturbations-induced) fixedrate at time te to absorb excessive Dpp concentration wereperformed in Reference 25. Extensive numerical simula-tions spanning a 6-dimensional parameter space showedthat less than 9% of gradients are of appropriate size andshape but with a mean robustness index (a numericalvalue to be defined in a later section as a measure of the

384 LANDER ET AL.

deviation of the aberrant gradient from the wild type)more than doubling the threshold value defined to beacceptably close to the wild-type gradient prior to the per-turbations. Most gradients generated are not biologicallyrealistic for cell differentiation (for reasons such as highreceptor occupancy throughout the wing imaginal discaway from its distal edge). These observations havebeen validated theoretically in References 17, 26, and 45.Adding negative feedback on receptor synthesis rate to amodest concentration of non-receptors was found in Refer-ence 25 to result in a slightly broader range of robust gra-dients that are biologically useful and robustness cannotbe attained with higher non-receptor synthesis rates(than the receptor synthesis rate) with any level of nega-tive feedback on receptor synthesis rate. Additional nega-tive feedback on the prescribed non-receptor synthesiswould only further degrade the aberrant gradient.

The theoretical results of Reference 26 on a Hill func-tion type feedback on receptor synthesis suggest that a spa-tially uniform feedback mechanism may avoid the effect ofshape distortion associated with spatially nonuniform feed-back. A (nonlocal) robustness-index based feedback mecha-nism has been developed in References 22 and 27 toprovide a spatially uniform feedback process. More detailsand support (including known examples) for such feedbackmechanisms will be deferred to a later section on our spe-cific nonlocal spatially uniform feedback. We note here onlythat the improvements associated with the new feedbackprocess applied to receptor synthesis rate (and a few otherbiological processes, such as the binding rate, and receptor-mediated degradation rate) were found insufficient forrobust signaling.22 However, the investigation enabled us touncover unexpected benefits of appropriate multi-feedbackmechanisms that are much more efficient in promotingrobust signaling gradients. In relation to the numerical sim-ulations of Reference 25, we report in this article one spe-cific application of the multi-feedback combination to showhow aberrancy induced positive feedback on non-receptorsynthesis rate may be combined with a similar negativefeedback on receptor synthesis for a very effective strategyfor promoting robust signaling in biologically realisticranges of system parameter values. We do this by examin-ing a set of models that are the counterparts of those inves-tigated in Reference 25 but now with our new spatiallyuniform feedback instrument instead of the Hill functiontype previously employed. One significant feature of ourapproach is that the new models admit explicit exact solu-tions for biologically realistic gradients so that our resultsare theoretically conclusive and do not rely on numericalsimulations. How non-receptors may or may not promoterobust signaling can be seen explicitly from the mathemati-cal expressions in terms of known functions for the signal-ing gradient concentration of the different models.

2 | A SIMPLE EXTRACELLULARMODEL OF DPP GRADIENTFORMATION

To understand better the results of numerical simula-tions of Reference 25, we re-examine the same threeapproaches to robust signaling there but now by way ofa spatially uniform feedback. For this purpose, we workwith the normalized form of the one-dimensional extra-cellular model of the Dpp gradient formation of Refer-ence 7. To simplify the analysis without sacrificing anyof the essential characteristics of the biological processesinvolved, we may take the wild-type Dpp synthesis rateVL(X, T) to be uniform in the direction along the bound-ary X = − Xmin between the anterior and posterior com-partments of the wing imaginal disc. Here, X is distancein direction perpendicular to the between-compartmentboundary with X spanning [−Xmin, Xmax], Xmax

being the edge of the posterior compartment. WithDpp synthesized at a uniform rate in a narrow strip−Xmin < X < 0, we idealize the synthesis rate by

VL X ,Tð Þ= !VLH −xð Þ ð1Þ

where x = X/Xmax. We also take the wild-type receptorsynthesis rate VR(X, T) to be uniform throughout the pos-terior compartment with a steady state receptor concen-tration of

R0 =VR X ,Tð Þ

kR=

!VR

kR=

UniformTkv synthesis rateTkvdegradation rate constant

ð2Þ

prior to the onset of Dpp synthesis at T = 0. Just as(the normalized) distance x in direction normal to thecompartment boundary measured in units of themaximum distal width Xmax of the posterior compart-ment, we also normalize the physical time T by set-ting t=DT=X2

max where D is the uniform diffusioncoefficient.

Normal development of wing imaginal disc and otherbiological organisms may be altered by an enhanced mor-phogen synthesis rate stimulated by sustained genetic orepigenetic changes (in contrast to a one time perturba-tion of an existing steady state), starting at some timete ≥ 0. For example, Dpp synthesis rate in Drosophilaimaginal disc has been shown to double when the ambi-ent temperature is increased by 5.9 !C (shown by S. Zhouwhile in A.D. Lander's Lab). At a state of low receptoroccupancy (LRO), basic models for signaling gradient for-mation would have the corresponding steady state aber-rant (abnormal) signaling ligand concentrationincreasing proportionately (see Equations (19) and (20)below) and its shape altered (and hence the cell fate at

LANDER ET AL. 385

each spatial location as well).7,16 Without the restrictionof low receptor occupancy, these and other models alsohave also the steady state aberrant signaling gradientmagnitude increased with synthesis rate, though not nec-essarily proportionately.7,16,17,26

Natural biological developments however are mostlyunaffected by sustained environmental perturbationsthat these models would have altered them. To investi-gate possible mechanisms for managing (down-regulat-ing) possible aberrant developments induced by geneticor epigenetic perturbations, we introduce an excess(amplification) factor e ≥ 1 and work with a more gen-eral ligand synthesis rate VL X ,Tð Þ= e!VLH −xð Þ insteadof Equation (1). The basic extracellular model for Dppgradient formation of Reference 7 then consists of the fol-lowing three normalized differential equations for thenormalized concentrations of free Dpp concentrationae(x, t), Dpp-Tkv complexes concentration (or signalingDpp gradient for short) be(x, t), and the unoccupiedTkv concentration re(x, t), all measured in units of thesteady state receptor concentration R0 introduced inEquation (2):

∂ae∂t

=∂2ae∂x2

−h0aere + f 0be + ve, ð3Þ

∂be∂t

= h0aere− f 0 + g0ð Þbe,∂re∂t

= vR−h0aere + f 0be−grre,

ð4Þ

where the quantities h0, g0, and f0 are (per unitmorphogen concentration) binding rate, receptor-mediated degradation rate and dissociation rate, allnormalized by D=X2

max . In the absence of feedback, thenormalized Dpp and Tkv synthesis rates, ve and vR, aregiven by

ve =eVL=R0

D=X2max

= evLH −xð Þ, vR =VR=R0

D=X2max

=kR

D=X2max

$ gr:

ð5Þ

where !vL is a dimensionless morphogen synthesis rate(prior to exogenous perturbations) and e is the ligandsynthesis amplification factor with e = 1 for thewild type.

The three differential equations are supplemented bythe boundary conditions

x= −xm :∂ae∂x

=0, x=1 : ae =0, ð6Þ

all for t > 0, and the initial conditions

t=0 : ae = be =0, re =1: ð7Þ

In addition to the killed end at x = 1, the sealed end con-dition at the boundary between the anterior and posteriorcompartments is a consequence of the idealized symme-try of the two compartments.

The initial-boundary value problem (IBVP) defined byEquations (3)-(7) and its modified forms have been ana-lyzed as mathematical models for ligand activities and tis-sue pattern formation in several of the references cited(eg, References 7, 25, and 26). Some basic results fromReference 7 are summarized below for comparison withthe new results on the effects of non-receptors to be ana-lyzed herein.

2.1 | Time independent steady state

Given that both the ligand and receptor synthesis ratesare time independent, it has been shown in Reference 7that the extracellular model system (3)-(7) has a uniquesteady state,

!ae xð Þ,!be xð Þ,!re xð Þ! "

= limt!∞

ae x, tð Þ,be x, tð Þ,re x, tð Þf g, ð8Þ

that is asymptotically stable with respect to small(one time) perturbations. With the three dependentvariables not changing with time in steady state, thegoverning IBVP may be reduced to the following well-posed two-point boundary value problem (BVP) for!ae xð Þ7:

!a00e −g0!ae

α0 + ζ0!ae+ e!vLH −xð Þ=0, ð9Þ

!a0e −xmð Þ=0, !ae 1ð Þ=0: ð10Þ

with

!be xð Þ=!ae xð Þ

α0 + ζ0!ae xð Þ, !re xð Þ= α0

α0 + ζ0!ae xð Þð11Þ

where

α0 =f 0 + g0h0

, ζ0 =g0gr: ð12Þ

We note again that the excess factor e is a constant forthe level of abnormality in the ligand synthesis rate withe = 1 for the wild-type development.

For the signaling gradient to induce a distinct biolog-ical tissue pattern, it should not be nearly uniform over a

386 LANDER ET AL.

significant spatial span of the solution domain (−xm, 1)as there would not be a pattern over that span. Forthis reason, the free morphogen concentration !ae xð Þassociated with a biologically realistic gradient systemcannot be so large that

ζ0!ae % α0, x<1ð Þ ð13Þ

or (with f0 & g0) !ae % gr=h0 away from the edge x = 1.For the unlikely event that the condition (13) should bemet, we would have

!be xð Þ=!ae xð Þ

α0 + ζ0!ae xð Þ’ 1ζ0

=grg0

uniform in x except for a boundary layer adjacent tothe edge x = 1. While ζ0!ae xð Þ=O α0ð Þ is not the onlyrequirement for a biologically realistic gradient system,we formulate it as a necessary condition for systemsworthy of examination.

Criterion 1 For a morphogen gradient system to indu-ce a biological meaningful pattern, the free morpho-gen concentration !ae must be O(α0/ζ0) = O(gr/h0) orsmaller.

2.2 | Low receptor occupancy

At the other extreme, when the free morphogen concen-tration !ae xð Þ is sufficiently low over the span of the solu-tion interval (0, 1) so that

ζ0!ae = g0!ae=gr & α0, ð14Þ

we may neglect terms involving ζ0!ae in Equations (9)-(11)to get an approximate set of solutions {Ae(x),Be(x),Re(x)}determined by the initial value problem (IVP)

A00e −μ20Ae + e!vLH −xð Þ=0, ð15Þ

A0e −xmð Þ=0, Ae 1ð Þ=0: ð16Þ

with

Be xð Þ= Ae xð Þα0

, Re xð Þ=1: ð17aÞ

and

μ20 =g0α0

=g0h0g0 + f 0

ð18Þ

The exact solution for Ae(x) ≡ eA1(x) obtained previouslyin Reference 7 is

with

!be xð Þ’Be xð Þ= eB1 xð Þ= eA1 xð Þα0

, !re xð Þ’ 1: ð20Þ

It would be natural to characterize a morphogen sys-tem with Equation (14) to be in a state of low receptoroccupancy (LRO) since there are few free ligand availableto occupy the signaling receptors. However, if we havealso μ0 % 1, the expression for Ae(x) in the signalingrange of 0 ≤ x < 1 is effectively a boundary layer adjacentto the edge of the ligand production region, steep nearx = 0 and dropping sharply to e!vL=μ20 (which is rathersmall) away from that boundary. Generally, the bound(signaling) morphogen gradient !be xð Þ’Be xð Þ shouldchange gradually if it is to lead to a distinct biological pat-tern. To limit our discussion to these biologically mean-ingful gradient systems, we adopt the following definitionfor a gradient system in a steady state of LRO:

Definition 2 A morphogen system is in a steady state oflow receptor occupancy (LRO) if the condition (14)is satisfied and μ0 = O(1).

With the adoption of this definition, we may then restrictour attention mainly to LRO systems that give rise to

A1 xð Þ=

!vLμ20

1−cosh μ0ð Þ

cosh μ0 1+ xmð Þð Þcosh μ0 x+ xmð Þð Þ

# $−xm ≤ x ≤ 0ð Þ

!vLμ20

sinh μ0xmð Þcosh μ0 1+ xmð Þð Þ

sinh μ0 1−xð Þð Þ 0≤ x ≤ 1ð Þ

8>>><

>>>:, ð19Þ

LANDER ET AL. 387

distinctive biological tissue patterns. For such systems,the bound and free ligand concentrations change onlygradually over their spatial span [0, Xmax]. For gradi-ent systems for which neither Equation (14) nor (13) ismet, the following condition provides a criterion for elim-inating a group of gradient systems that is not biologi-cally realistic and not of interest herein:

Criterion 3 A gradient system is not biologically mean-ingful if μ0 % 1.

While Criterion 1 eliminates signaling gradients that arepretty much flat and with signaling receptors saturated awayfrom the edge x = 1, Criterion 3 eliminates signaling gradi-ents that are also pretty much flat but with signaling recep-tors sparsely occupied away from ligand synthesis region.

2.3 | Root-mean-square signalingdifferential

We wish to make use of the extracellular model summa-rized in the preceding subsections to gain more insightinto the results from numerical simulations obtained inReference 25 and to investigate more effective feedbackmechanisms to ensure robust development. We do this byworking with a spatially uniform feedback to complementthe conventional Hill function approach in modeling feed-back processes. For this purpose, we need to have a quan-titative measure of robustness that quantifies succinctlydeviation from the wild-type development. One suchmeasure, designated as the root-mean-square signaling dif-ferential robustness index (henceforth the Rb robustnessindex for short, or simply robustness index when there isno ambiguity) used in Reference 16, is given below andapplied to illustrate its effectiveness in measuring the aber-rancy of a signaling gradient. This index is simpler toanalyze compared to the root-mean-square displacementdifferential robustness index Rx (previously denoted by L inReference 25) which was also used earlier in Reference 16and will be examined in later sections of this article.

The (signal) robustness index Rb is the (normalized)root mean square of the deviation of the aberrant signalinggradient be(x, t) from wild-type signaling gradient b1(x, t):

Rb tð Þ= 1bh−bℓ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

xℓ−xhð Þ2ðxℓ

xhbe x, tð Þ−b1 x, tð Þ½ (2dx

s

ð21Þ

where 0 ≤ bℓ(t) < bh(t) ≤ b1(−xm, t) and −xm ≤ xh< xℓ ≤ 1. The quantities xℓ, xh, bℓ, and bh may be chosenaway from the extremities to minimize the exaggeratedeffects of outliers. For a system in steady state with

!b1 xð Þ= limt!∞

b1 x, tð Þ, !be xð Þ= limt!∞

be x, tð Þ, ð22Þ

Rb(t) tends to a constant !Rb with

!Rb = limt!∞

Rb tð Þ= 1bh−bℓ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

xℓ−xhð Þ2ðxℓ

xh

!be xð Þ−!b1 xð Þ' (2dx

s

:

ð23Þ

In subsequent developments, we set xh = 0 since theregion of ligand synthesis [−xm, 0) is not expected to contrib-ute significantly to signaling. We also take xℓ = 1 so thatbℓ(1, t) = b1(1, t) = 0. For the case of low receptor occupancy,we take bh to be B1(0), the explicit LRO approximation ofthe steady state wild-type signaling gradient concentrationvalue !b1 0ð Þ, known from Equations (19) and (20) to be

bh =B1 0ð Þ=!vLα0μ20

sinh μ0xmð Þsinh μ0ð Þcosh μ0 1+ xmð Þð Þ

’ !b1 0ð Þ: ð24Þ

Having !be xð Þ and !b1 xð Þ (by any numerical software forBVP of ODE), it is straightforward to evaluate the inte-gral (23) to obtain !Rb to see whether or not the aberrancyof the signaling gradient (when distorted by exogenousperturbations) is still acceptable.

2.4 | Approximate robustness index forLRO state

For a morphogen system in a state of LRO, we havefrom Equations (19) and (20) the following approxi-mate steady state solutions for the distorted signalinggradients, Be(x), of the (environmentally or genetically)perturbed system:

Be xð Þ ) eB1 xð Þ= evLα0μ20

sinh μ0xmð Þsinh μ0 1−xð Þð Þcosh μ0 1+ xmð Þð Þ

, 0≤ x ≤ 1ð Þ

ð25Þ

where μ20 = g0=α0 ’ h0 since f0& g0 for our model of thewing imaginal disc.7 The parameter e is the excess(amplification) factor of the ligand synthesis rate. Thenthe LRO approximation of !Rb (with xℓ = 1, xh = 0), den-oted by ρe, is given by

!Rb ) ρe =e−1

sinh μ0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0sinh μ0 1−xð Þð Þ½ (2dx

s

=e−1

sinh μ0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

sinh 2μ0ð Þ2μ0

−1) *s

$ e−1ð Þγ μ0ð Þ:

ð26Þ

388 LANDER ET AL.

To be concrete and to make use of the finding ofZhou on the effect of a 5.9 !C temperature change, weare mainly concerned with the empirically observed caseof e = 2 in the subsequent development.

For a gradient system with g0 = 0.2, f0 = 0.001, gr = 1,h0 = 10, xm = 0.1, and !vL =0:05 (corresponding to!VL =0:002 μM, !VR =0:04 μM, D = 10−7 cm2/s, Xmax =0.01 cm) with β=!vL=g0 = 0:25 in table 2 of Reference 7,the steady state is in LRO state. For this case, the approxi-mate solution for !Rb given by Equation (26) is 0.3938…while the actual solution for !Rb computed from anaccurate numerical solution for the BVP for !a2 xð Þ gives0.3943… for a percentage error of less than 0.01%. If ligandsynthesis rate is increased 20 times to !VL =0:04 μM, thepercentage error of the low receptor occupancy approxi-mation is still less than 1%. These comparisons serveto validate the numerical simulation code developedfor exact numerical solutions of our model. (Consistentwith the subsequent ease of analysis for the LRO, wehave adopted the simplifying approximation α0’ g0/h0in the computation for both !Rb and ρe in this articlesince f0& g0.)

Our main interest however is in the use of the robust-ness index to induce an appropriate feedback mechanismfor attaining robustness of signaling morphogen gradi-ents. When an enhanced ligand system is not in a state oflow receptor occupancy, the use of the approximate sig-naling robustness index based on the approximate solu-tion (25) may not be sufficiently accurate. For thesecases, numerical solutions of !be xð Þ and the correspondingvalue for !Rb should be used instead of Be(x) and ρe. Ifneeded, their calculations by available mathematical soft-ware such Mathematica, Matlab or Maple should bestraightforward.

3 | FEEDBACK ON RECEPTORSYNTHESIS RATE

Excessive ligand concentration is known to down-regulate its own signaling receptor synthesis. In particu-lar, Decapentaplegic (Dpp) represses the synthesis ofits own receptor Tkv.4 Another example is Wingless (Wg)repressing its signaling receptor DFz2.46 The down-regulation of Tkv by aberrant signaling Dpp gradient wasmodeled by a negative feedback of the Hill function typein Reference 25 and was found to be ineffective as amechanism for promoting robust development of the wingimaginal disc. A theoretical analysis of the model inReference 26 confirms the results of the numerical experi-ment and shows that the spatially nonuniform feedbackdistorts the shape of the output gradient as the feedbackmechanism works to reduce its aberrancy. The observation

suggests that a spatially uniform feedback mechanismmay be more effective for ensuring robustness. We con-sider in Reference 22 the originally (normalized) spatiallyuniform receptor synthesis rate !vR being down-regulatedto vR(t) by a negative feedback factor κ2(t) that is a func-tion of the signaling robustness index Rb(t) in the form

vR tð Þ= κ2 tð Þ!vR =!vR

1+ c Rb t−τð Þ½ (mð27Þ

where the parameter τ corresponds to a possible timedelay and {c, m} are two parameters to be chosen forappropriate feedback strength and sensitivity, respec-tively, similar to those for a Hill's function.

It should be noted that the feedback process in Equa-tion (27) at any location does not depend on the aberrancyof the signaling gradient at that location, only on an aver-age measure Rb(t) of the excess over the span of the wingdisc. Since it is not sensitive to the local environment ofindividual cell, it is less likely to contribute to the shapedistortion of the resulting gradients. Such a spatially uni-form feedback mechanism obviously requires some disc-wide cooperation among cells. There are at least two possi-ble examples of biological mechanisms that could createsuch spatial coordination, particularly in the wing disc.

One of these was described in the two papers by Bar-kai et al.47,48 These papers describe an “expander-repressor scaling mechanism” in which Dpp represses anexpander which then diffuses back into the wing disc andexpands the Dpp gradient. In the model, the expanderfreely diffuses everywhere, which is why it is able toadjust Dpp decay uniformly across the disc. It is similarto Equation (27) in that some aspect of Dpp gradientrobustness is being controlled by Dpp through a mecha-nism that is spatially uniform and is an example of hownon-spatial feedback might be triggered by changes to amorphogen gradient.

The other is the work of Yu et al in Reference 49where the Fat/Dachsous pathway coordinates Dpp sig-naling over wide spatial ranges for cell growth. This is acompletely different mechanism from the one developedin References 47 and 48; but it has the same feature thatcells end up cooperating over large spatial scales. Thework shows another mechanism that allows an entiredisc to respond to local perturbations.

Together with additional comments on such a non-local feedback mechanism in Reference 22, the specificexamples above provide concrete evidence supporting thepossibility of spatially uniform non-local feedback pro-cesses such as Equation (27) and those to be introducedin subsequent developments. For the investigation of theeffects of our particular type of negative feedback on thereceptor synthesis rate, we are interested in the modified

LANDER ET AL. 389

signaling gradient (starting at t = 0) and the correspondingrobustness index of the IBVP (3)-(7) but now for anenhanced ligand synthesis rate

vL = e!vLH −xð Þ ð28Þ

with an excess factor e > 1 and a down-regulated receptorsynthesis rate given by Equation (27). In the presence ofthe feedback, the three aberrant gradients of the new IBVPare to be denoted by {av(x, t), bv(x, t), rv(x, t)} ≡ {ae(x, t; c),be(x, t; c), rv(x, t; c)} which reduce to {ae(x, t), be(x, t), re(x,t)} of the model without feedback (when c = 0). Thecorresponding robustness index is determined by

Rb tð Þ= 1bh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0bv x, tð Þ−b1 x, tð Þ½ (2dx

s

: ð29Þ

Unlike the situation in Equation (21), bv(x, t) = be(x, t; c)now depends on Rb(t). As a consequence, Equation (29)is an integral equation to be solved for the unknownRb(t) concurrently with the solution of the IBVP (3)-(7)with !vR replaced by vR tð Þ= κ2 tð Þ!vR. Such solutionshave been obtained in References 20 and 21 and willnot be pursued herein. Instead, we will focus on thecorresponding steady state behavior with low receptoroccupancy.

3.1 | Time independent steady state withfeedback

It has been shown in Reference 7 that the extracellularmodel system without feedback has a unique steady state.We are interested here also in the corresponding steady statefor the case with the spatially uniform non-local feedbackon the signaling receptor synthesis rate !VR of the type char-acterized by Equation (27). We denote the correspondingtime independent steady state aberrant gradients by!av xð Þ,!bv xð Þ,!rv xð Þ

! "$ !ae x;cð Þ,!be x;cð Þ,!re x;cð Þ! "

and thesteady state robustness index !Rb determined by

!Rb = limt!∞

Rb tð Þ= 1bh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0

!bv xð Þ−!b1 xð Þ' (2dx

s

ð30Þ

with bh appropriately taken to be B1(0), the LRO approxi-mation for !b1 0;cð Þ

' (c=0 =

!be 0;0ð Þ' (

e=1 . (Note that !Rb ispositive unless c = 0 and e = 1.)

With ∂( )/∂t = 0, the governing equations and bound-ary conditions for our extracellular model with feedbackcan again be reduced to a BVP for !av alone:

!a00v −!κ2g0!av

α0 + ζ0!av+ e!vLH −xð Þ=0, ð31Þ

!a0v −xmð Þ=0, !av 1ð Þ=0: ð32Þ

where α0 and ζ0 as defined in Equation (12) and

!κ2 !Rbð Þ= 11+ c!Rm

b

: ð33Þ

The corresponding signaling ligand and unoccupiedreceptor concentrations are given in terms of !av by

!bv xð Þ=!κ2!av xð Þ

α0 + ζ0!av xð Þ, !rv xð Þ=

!κ2α0α0 + ζ0!av xð Þ

, ð34Þ

respectively. The solution for !bv xð Þ with !Rb as anunknown parameter is then used in Equation (30) for thedetermination of !Rb.

Existence of a unique, non-negative, monotonedecreasing solution for Equations (31) and (32) can beproved by the method used in Reference 7. Sample solu-tions of the BVP and the signaling gradient !bv xð Þ=!be x;cð Þ for c>0 (with !bv xð Þ

' (c=0 =

!be x;0ð Þ= !be xð Þ withoutfeedback) have been calculated and analyzed in Refer-ence 22. Here, we complement these results by showingthe existence and uniqueness of a positive robustnessindex !Rb for the particular feedback problem. Themethod may be used to establish similar results for prob-lems that include the effects of non-receptors in latersections.

To be concrete, we take xℓ = 1, xh = 0, and bℓ = 0henceforth so that

!Rb =1bh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0

!be x;cð Þ−!b1 xð Þ' (2dx

s

$C !Rbð Þ ð35Þ

As indicated previously, we take bh to be given by Equa-tion (24). We now work with Equation (35) to show firstthat for a fixed positive c, !be x;cð Þ is a decreasing functionof !Rb and then use the result in

dCd!Rb

=1

b2hC

ð1

0

!be x;cð Þ−!b1 xð Þ' (∂!be x;cð Þ

∂!Rbdx, ð36Þ

to establish the existence, uniqueness and positivity of!Rb: For simplicity, we do this for the m = 1 case.

Upon differentiating all relations in the BVP for!av xð Þ= !ae x;cð Þ partially with respect to !Rb, we obtain

390 LANDER ET AL.

−w00 +!κ2g0

α0 + ζ0!avð Þ2w−

c!κ4g0!avα0 + ζ0!av

=0, ð37Þ

w0 −xm;cð Þ=0, w 1;cð Þ=0 ð38Þ

where

w x;cð Þ= −∂!ae x;cð Þ∂!Rb

= −∂!av xð Þ∂!Rb

:

Clearly, wℓ(x; c) ≡ 0 is a lower solution of the BVP for w(x; c) given

−w00ℓ +

!κ2g0α0 + ζ0!avð Þ2

wℓ−c!κ4g0!avα0 + ζ0!av

= −c!κ4!av

α0 + ζ0!av≤ 0

and the conditions in Equation (38) are also satisfied bywℓ(x; c) ≡ 0. As an upper solution, we take

wu x;cð Þ= α0grg0

e!vL xm +12

) *−xmx−

12x2

# $$ α0gr

g0au xð Þ

so that the boundary conditions for w(x) are satisfied withwu(x; c) ≥ 0 for all x in [−xm, 1]. The function au(x), there-scaled wu(x; c), was shown in Reference 7 to be anupper solution for !av xð Þ so that au xð Þ≥!av xð Þ. With ζ0 = O(10−1) while c,1 + c!Rb and 1+ ζ0!av=α0 are O(1) quantities,we have

−w00u +

!κ2g0α0 + ζ0!avð Þ2

wu−c!κ4g0avα0 + ζ0!av

=

α0grg0

e!vL +!κ2gr

α0 + ζ0!av

au1+ ζ0!av=α0

−cζ0!av1+ c!Rb

# $≥0:

Then the monotone method in References 50-52 impliesthat w(x; c) exists, is unique and non-negative so that

−w x;cð Þ≤ ∂!ae x;cð Þ∂!Rb

=∂!av xð Þ∂!Rb

) *≤ 0 ð39Þ

The development above leads to the following lemmaon the non-positivity of the marginal value ∂~be x;cð Þ=∂!Rb:

Lemma 4 ∂!be x;cð Þ=∂!Rb ≤ 0.

Proof Upon differentiating the expression for!bv xð Þ= !be x;cð Þ in Equation (34) partially withrespect to !Rb, we obtain

∂!bv xð Þ∂!Rb

=!κ2 Rbð Þα0α0 + ζ0!avð Þ2

∂!av xð Þ∂!Rb

−c!κ4!av

α0 + ζ0!av≤ 0

in view of the second inequality of Equation (39).

Proposition 5 A positive solution of !Rb =C !Rbð Þ existsand is unique.

Proof Together with !be x;cð Þ>0 , Lemma 4 impliesdC=d!Rb ≤ 0 as long as !be x;cð Þ> !b1 xð Þ (see Equa-tion (36)). It follows that a solution exists for!Rb =C !Rbð Þ . It is unique since C !Rbð Þ is monotonedecreasing. It is positive since C !Rbð Þ is nonnegativeso that !Rb is bounded below for !Rb≥0.

3.2 | Low receptor occupancy

The LRO approximation for morphogen systems hasbeen found useful for an understanding of the effects ofvarious feedback mechanisms. The steady state approxi-mation for the present feedback process has beenobtained in Reference 22. The results are summarizedbelow for reference later in the study of the effects ofmulti-feedback system involving non-receptors on robust-ness. In a state of low receptor occupancy prior to andafter ligand synthesis enhancement so that

ζ0!av &α0 ð40Þ

(including the special case where c = 0 and e = 1 so that!av xð Þ reduces to !a1 xð Þ= !ae x,0ð Þ½ (e=1 ), !av xð Þ may beapproximated by eAv(x) with

Av xð Þ=

!vLμ2c

1−cosh μcð Þ

cosh μc 1+ xmð Þð Þcosh μc x+ xmð Þð Þ

# $−xm ≤ x ≤ 0ð Þ

!vLμ2c

sinh μcxmð Þcosh μc 1+ xmð Þð Þ

sinh μc 1−xð Þð Þ 0≤ x ≤ 1ð Þ

8>>><

>>>:, ð41Þ

LANDER ET AL. 391

where

μ2c =!κ2 ρcð Þ g0α0

: ð42Þ

and ρc is the LRO approximation of !Rb calculated fromEquation (30) using eAv(x) and A1(x,0) for !av xð Þ and!a1 xð Þ= !ae x,0ð Þ½ (e=1 , respectively (see Reference 22). TheLOR solution eAv(x) is expected to be an accurate approx-imation of the exact solution !av xð Þ and reduces to theLRO wild-type ligand concentration A1(x) when c = 0and e = 1.

The corresponding LOR signaling gradient and freereceptor concentration are given by

!bv xð Þ= !be x,cð Þ’ eBv xð Þ= eAv xð Þα0

, !rv xð Þ’ !κ2 ρcð Þ ð43Þ

with

Bv xð Þ=!vLg0

sinh μcxmð Þcosh μc 1+ xmð Þð Þ

sinh μc 1−xð Þð Þ 0≤ x ≤ 1ð Þ:

ð44Þ

It should be noted that the dissociation rate constant isusually much smaller than the degradation rate con-stant with f0/g0 = O(10−2). To simplify our discussion,we have adopted the simplifying approximationα0 = (f0 + g0)/h0 ’ g0/h0 so that

g0α0

’ h0 $ μ20, μ2c =μ20

1+ cρmcð45Þ

as in Reference 22.For e > 1, the yet unknown LOR robustness index ρc

is to be determined by the LRO approximation ofEquation (30)

!Rb ) ρc =1

B1 0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0eBv xð Þ−B1 xð Þ½ (2dx

s

$Cv ρcð Þ: ð46Þ

Given Equations (19), (20), and (44), the right handside depends on ρc through Bv(x); hence Equation (46) isan equation for ρc ’ !Rbð Þ:

f c ρcð Þ=0

where

f c ρcð Þ$ 2sinh2 μ0ð Þρ2c −e2csinh 2μcð Þ

2μc−1

) *−

sinh 2μ0ð Þ2μ0

−1) *

+2ecsinh μc + μ0ð Þ

μc+ μ0−sinh μc−μ0ð Þ

μc−μ0

# $,

ð47Þ

with

ec = esinh μcxmð Þcosh μ0 1+ xmð Þð Þsinh μ0xmð Þcoshðμc 1+ xmð Þ

, μ2c =h0

1+ cρmc:

ð48Þ

Even without an explicit solution for fc(ρc) = 0, we seefrom Equation (46) that ρc is necessarily positive whene > 1 and therewith 0< !κ2 ρcð Þ<1: It follows from

eBv 0ð Þ= e!vLg0

sinh μcxmð Þcosh μc 1+ xmð Þð Þ

sinh μcð Þ

that the order of magnitude of Be(0, c) = eBv(0) is notchanged in any significant way by the presence of theadopted feedback. With μ20 ’ h0 =O 10ð Þ, μ2c = !κ2 ρcð Þμ20and 0< xm& 1, we can work with the approximateexpression estimate

Bv 0ð ÞB1 0ð Þ

=sinh μcð Þsinh μ0ð Þ

sinh μcxmð Þsinh μ0xmð Þ

cosh μ0 1+ xmð Þð Þcosh μc 1+ xmð Þð Þ

=1−e−2μcxm + o e−2μcxmð Þ1−e−2μ0xm + o e−2μ0xmð Þ

’ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ cρmc

pð49Þ

to get

eBv 0ð ÞB1 0ð Þ

=Oeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1+ cρmcp

!

:

with ρc < 0.2 for a robust gradient. Note that the ratio issmallest when the positive integer m is 1.

In addition, the negative feedback (27) also leads to aless convex modified ectopic signaling gradient sinceμc < μ0 whenever ρc > 0. We summarize the developmentabove by the following observation:

Conclusion 6 When both the wild-type and aberrant gra-dients are in a state of LRO, the negative feedbackmechanism (27) on receptor synthesis rate isnot particularly effective for promoting robustdevelopment.

392 LANDER ET AL.

3.3 | Numerical solution for thegeneral case

In order to confirm the usefulness of the LOR solutionfor the problem, we have re-configured the integro-differential equation problem for !av as a BVP for a systemof ODE and solved it by any BVP solver (eg, bvp4c inMatlab). Numerical results are reported below for a sys-tem characterized by the parameter values shown inTable 1 with m = 1.

The biological implication of the resulting robustnessindex is not particularly gratifying. The values of ρc and !Rb

shown in Table 1 differ by less than 1% for all five valuesof c. Their values for c = 1 are well above the acceptablelevel of 0.2 for a robust signaling gradient, a rathermodest requirement set arbitrarily in Reference 25 forrobustness. This is hardly surprising given the estimatefor the amplitude of the explicit solution eBv(0) for theLRO approximation of !bv 0ð Þ in Equation (49) being inde-pendent of feedback. Comparing the accurate numericalsolution for !b1 0ð Þ and !be 0,0ð Þ with !bv 0ð Þ= !be 0,cð Þreported in Table 1 shows that the latter is much closerto !be 0ð Þ than !b1 0ð Þ. More specifically, !be 0ð Þ and !bv 0ð Þ areroughly double the magnitude of !b1 0ð Þ , confirming theineffectiveness of the feedback !κ2 Rbð Þ for c = 1.

The LOR solution eBv(x) also shows that increasingthe value of the parameter c to larger than 1 would fur-ther distort the shape of the gradient and ameliorate theamplitude reduction of the ectopic gradient through theterms involving hyperbolic functions with increasinglysmaller μc. Accurate numerical solutions for the generalcase (not in a state of LRO) in Table 1 are qualitativelysimilar to the LRO results with the robustness indexdecreasing rather gradually as c increases. Figure 1 showsgraphs of !be x,cð Þ for c = 0, 1, 2 and 4 for comparison with

FIGURE 1 Spatially uniform negative feedback on receptorsynthesis rateT

ABLE

1So

mesamplesolutio

nsX

max=0:01

cm,

Xmin=0:001cm

,k o

nR0=

0:01=s=μ

M,

m=1,

k deg=2×10

−4 =s,

k R=0:001=s,

k off=10

−6 =s,

e=2,

D=10

−7cm

2 =s,

! VL=0:002μM

=s,

! VR=0:04

μM=s

c! Rb

ρ c! b e

0;c

ðÞ

eBv(0)

! b 10ðÞ

B1(0)

00.3943

0.3938

0.1154

0.1169

0.0580

0.0584

0.5

0.3750

0.3736

0.1080

0.1096

0.0580

0.0584

10.3573

0.3565

0.1022

0.1062

0.0580

0.0584

20.3284

0.3284

0.0941

0.0959

0.0580

0.0584

40.2862

0.2872

0.0839

0.0857

0.0580

0.0584

LANDER ET AL. 393

!b1 xð Þ to further illustrate these observations. Note thatfor all the graphs in the figure, the labels are generically!be x;c,Z,η,σð Þ with Z = η = σ = 0 for Figure 1 (since theseparameters do not appear until later sections) and!be x;c,0,0,0ð Þ$ !be x,cð Þ:

It is necessary then to look elsewhere for a moreeffective feedback instrument to attain robust develop-ment with respect to an aberrant Dpp synthesis rate.There are a number of different such mechanisms avail-able for this purpose.22 In the next few sections, we willfocus on only one of these, namely, the role of non-receptors on robust development to gain insight to thefindings for model 3 in Reference 25 and to help developa more effective feedback strategy for promoting robustsignaling gradients, hence robust biological development.

As we routinely work with normalized quantities,we give below for subsequent references the normalizedparameter values corresponding to the actual biologicalparameter values listed under Table 1 above to be used forcomputing various solutions:

g0 = 0:2, gr =1:0, f 0 = 0:001, xm =0:1, vL =0:05 ð50Þ

The normalized value of receptor synthesis rate does notappear explicitly in the dimensionless BVP for the steadystate solution, (31)-(34). It is involved in various normal-ized quantities such as vL and will not be given here.

4 | EFFECTS OF NON-RECEPTORS

The existence of inhibiting non-receptors and the associ-ated feedback process are well documented for the BMPfamily ligands that includes the Dpp ligand of interesthere (see References 19 and 31 and elsewhere). Knownnon-receptor type inhibitors include noggin, chordin,dally, follistatin, sog and various heparan sulfate proteo-glycans. They are ubiquitous during wing imaginal discand other biological developments (see References 30, 32,33, 36-38, 40, and 41 and references cited earlier). For thepurpose of establishing robust signaling Dpp gradients,we are interested in the effects of relevant non-receptorsas an inhibiting agent on such gradients. The impact ofnon-receptors on the (time independent) steady state of asignaling gradient has been investigated theoretically byanalysis and numerical simulations in References 17, 18,25, 44, and 45. To the extent that inhibitors for promotingrobust gradients are expected to be induced by sustainedexogenous perturbations (as it would be a part of thewild-type development otherwise), the introduction ofnon-receptors in these models may be interpreted as anon-receptor synthesis rate of the form

VN = !VNH t− teð Þ ð51Þ

where te > 0 is the instant of the onset of genetic or epige-netic perturbations. In this section, we examine the con-sequences of non-receptors generated in this wayprincipally to delineate similar results obtained in Refer-ence 25 for Equation (51) in the presence of a negativefeedback on receptor synthesis rate. The usefulness ofnon-receptors for promoting signaling gradient robust-ness will be investigated for a robustness index inducedfeedback mechanism on non-receptor synthesis rate inthe next section.

To examine how the introduction of non-receptorsaffects the robustness of the signaling gradient as mea-sured by the robustness index, we take as a referencelevel of non-receptor concentration N0 = !VN=jN with

Z=N0

R0=

!VN=jN!VR=kR

, !vN =X2

max

D

!VN

N0

# $=

jND=X2

maxð52Þ

where jN is the degradation rate constant for the unoccu-pied non-receptors. Similar to signaling receptors,(normalized) free (cell surface bound) non-receptor con-centration n(x, t) is also bound reversibly to Dpp ligand(to form normalized bound non-receptors of concentra-tion c(x, t)),

c,nf g=1N0

LN½ (, N½ (f g, ð53Þ

with a normalized binding rate constant h1a (for bindingbetween Dpp and non-receptors of concentration [N]),non-receptor-mediated degradation rate constant g1 (fordegradation of Dpp-non-receptor complexes [LN]), disso-ciation rate constant f1 (for dissociation rate of Dpp-non-receptor complexes) and free non-receptor degradationrate constant gn (for degradation of unoccupied non-receptors):

h1,g1, f 1,gnf g= X2max

DjonR0, jdeg, joff , jN

n o: ð54Þ

with f1 & g1 so that α1 ’ g1/h1.In terms of these normalized quantities, we have the

following IBVP for the five normalized unknowns a, b, r,c, and n45:

∂ae∂t

=∂2ae∂x2

−h0aere + f 0be−Zh1aene +Zf 1ce

+!vLH −xð Þ 1+ e−1ð ÞH t− teð Þf g,ð55Þ

394 LANDER ET AL.

∂be∂t

= h0aere− f 0 + g0ð Þbe,∂re∂t

=!vR−h0aere + f 0be−grre,

ð56Þ

∂ce∂t

= h1aene− f 1 + g1ð Þce,

∂ne∂t

=!vNH t− teð Þ−h1aene + f 1ce−gnne,ð57Þ

with the end conditions

x= −xm :∂ae∂x

=0, x=1 : ae =0, ð58Þ

for t > 0, and (for te > 0) the initial conditions

t=0 : ae = be = ce =ne =0, re =1 −xm < x <1ð Þ:ð59Þ

As before, the parameter e ≥ 1 is a measure of theexcess (amplification) of the enhanced Dpp synthesisrate induced by a sustained exogenous perturbationinitiated at te > 0. In this section, our first goal is toinvestigate whether the presence of a sufficiently highnon-receptor synthesis rate may make the eventual sig-naling morphogen gradient [LR] insensitive to anenhanced Dpp synthesis rate. Since we are interestedin the steady state behavior for large t, the initial condi-tions (59) will not have a substantive role in ouranalysis.

4.1 | Time-independent synthesis rates

When all synthesis rates for ligands, receptors and non-receptors are time-invariant, the solution of the new five-component model is expected to tend to a steady state den-oted by !aZ xð Þ,!bZ xð Þ,!cZ xð Þ,!rZ xð Þ,!nZ xð Þ

! "(as abbreviations

for !ae x;Zð Þ,…,!ne x;Zð Þf g when appropriate). For thissteady state solution, we have ∂( )/∂t = 0 so that the lastfour equations in (56) and (57) can be solved for!bZ xð Þ,!cZ xð Þ,!rZ xð Þ and !nZ xð Þ in terms of !aZ xð Þ to getEquation (11) with the subscript e replaced by Z and

!cZ xð Þ=!aZ xð Þ

α1 + ζ1!aZ xð Þ ,!nZ xð Þ= α1

α1 + ζ1!aZ xð Þ , ð60Þ

with

α1 =f 1 + g1h1

, ζ0,ζ1! "

=kdegkR

,jdegjN

# $=

g0gr,g1gn

# $: ð61Þ

The parameters kdeg and jdeg are the receptor- andnon-receptor-mediated degradation rate constant,respectively, of bound Dpp complexes. The results arethen used to obtain from the steady state version ofEquation (55),

!a00Z−h0!aZ!rZ + f 0!bZ−h1Z!aZ!nZ + f 1Z!cZ + e!vLH −xð Þ=0,

ð62Þ

and Equation (58) a BVP for !aZ xð Þ= !ae x;Zð Þ alone:

!a00Z−g0!aZ

α0 + ζ0!aZ−

Zg1!aZα1 + ζ1!aZ

+ e!vLH −xð Þ=0, ð63Þ

!a0Z −xmð Þ=0, !aZ 1ð Þ=0: ð64Þ

Evidently, !aZ xð Þ varies with Z (and occasionally writtenas !ae x;Zð Þ) which reduces to !ae xð Þ of the basic model forZ = 0, i.e., !ae xð Þ= !aZ xð Þ½ (Z=0 = !ae x;0ð Þ . Even if e = 1,!a1 x;Zð Þ is different from the solution !a1 xð Þ= !a1 x;0ð Þ, thefree ligand concentration in the wild-type developmentwithout non-receptors (corresponding to te = ∞).

4.2 | Low receptor and non-receptoroccupancy (LRNO)

For the signaling gradient to provide positional informa-tion that differentiates cell fates, the normalized concen-tration b = [LR]/R0 should not be nearly uniform (with asteep gradient adjacent to the absorbing edge or theligand synthesis region). Positional indifference exceptfor a steep gradient near the imaginal disc edge is notlikely to occur if free ligand concentration is sufficientlylow so that α0 + ζ0!aZ ’ α0 and α1 + ζ1!aZ ’ α1 . A gradientsystem is said to be in a state of low receptor and non-receptor occupancy (LRNO) if

α0 + ζ0!aZ ’ α0, α1 + ζ1!aZ ’ α1, 1 < μ0 < μZ =O 1ð Þ:ð65Þ

The gradient system is biologically differentiating if it is ina state of LRNO.

When the system is in a state of LRNO, the ODE (63)can be linearized with the corresponding approximatesolution denoted by {eAZ(x), eBZ(x), eCZ(x), eNZ(x),eRZ(x)}. The relevant BVP can be further simplified to!aZ xð Þ’ eAZ xð Þ:

A00Z−μ2ZAZ +!vLH −xð Þ=0, A0

Z −xmð Þ=AZ xð Þ=0 ð66Þ

LANDER ET AL. 395

where

μ2Z =g0α0

+Zg1α1

’ h0 +Zh1 ð67Þ

Its solution has been obtained in References 25 and 45and elsewhere. With f0 & g0 and f1 & g1, we neglect(as in Reference 22) the effect of the dissociation ratesand write AZ(x) as

with μ20 ’ h0, Zμ21 ’Zh1 and

μ2Z = μ20 +Zμ21 ’ h0 + h1Z: ð69Þ

The corresponding signaling gradient is !bZ xð Þ’ eBZ xð Þ with

BZ xð Þ= h0!vLg0μ

2Z

sinh μZxmð Þcosh μZ 1+ xmð Þð ÞÞ

sinh μZ 1−xð Þð Þ x≥0ð Þ

ð70Þ

(keeping in mind α0 = (g0 + f0)/h0 ’ g0/h0) and

BZ 0ð Þ= AZ 0ð Þα0

=h0!vLg0μ

2Z

sinh μZxmð Þsinh μZð ÞÞcosh μZ 1+ xmð Þð Þ

ð71Þ

Note that BZ(x) depends on Z through μZ.For h1 ≳ h0 (>g0) so that μ2Z = μ20 > 1, we have

sinh μkxmð Þsinh μkð ÞÞcosh μk 1+ xmð Þð Þ

) 12

1−e−2μkxm+ ,

k=0,Zð Þ

and

eBZ 0ð ÞB1 0ð Þ

’ e1+ h1Z=h0

1−e−2μZxm

1−e−2μ0xm,

μZxm = μ0xmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ h1Z=h0

p:

ð72Þ

where B1 0ð Þ= BZ 0ð Þ½ (Z=0 ’ !b1 0;0ð Þ+ ,

. For h1 = h0 andZ = O(1), we have (with 0< xm& 1)

!bZ 0ð Þ’ eBZ 0ð Þ’ e1+Z

!vLg0, ð73Þ

indicating that the amplitude of the aberrant signalingmorphogen concentration (at x = 0) is reduced (approxi-mately) to the wild-type concentration with Z = 1 for theempirically observable case of e = 2 (the amplification forDpp synthesis rate associated with an increase of 5.9 !Cin ambient temperature). However, this does not meanthat the aberrant gradient is reduced to the wild type

since the shape parameter μ2Z = h0 1+Zð Þ would be 2h0 sothat the slope and convexity of the aberrant signaling gra-dient would be changed substantially.

More generally, the aberrant concentration !bZ 0ð Þ’eBZ 0ð Þmay be kept to the wild-type level with a sufficientlyhigh non-receptor synthesis rate so that 1+h1Z/h0≥ e.However, for a given h1/h0 ratio, the required level ofZ increases with e resulting in at least two effects that causea distortion of signaling gradient shape and thereby workagainst the desired reduction of aberrancy of the signalinggradient. First, the ratio 1−e−2μZxmð Þ= 1−e−2μ0xmð Þ is sig-nificantly larger than 1 for larger Z and a larger Z isneeded to reduce the amplitude of the gradient (thanthat for 1 + h1Z/h0 = e). Second, the aberrant gradientshape would be distorted even more severely by a largerZ since the gradient shape parameter μ2Z increases line-arly with Z. Since the quantitative effects from a particu-lar non-receptor synthesis rate can only be seen fromthe corresponding value of the (LRNO approximation ofthe) robustness index, denoted by ρZ, the dependenceof ρZ on Z will be calculated in the next subsection.Here, we settle for the following relatively conservativeobservation:

Conclusion 7 For gradient systems in a steady state of lowreceptor and non-receptor occupancy with h1/h0 = O(1), the amplitude of their aberrant signaling gradientinduced by the empirically observed synthesis rateamplification factor of e = 2 could be kept close to thewild-type amplitude (around x = 0) by a moderatenon-receptor synthesis rate of Z ’ 1 (as shown by theexact [numerical] solutions without the LRNO approx-imation in Figure 2).

AZ xð Þ=

!vLμ2Z

1−cosh μZð Þ

cosh μZ 1+ xmð Þð Þcosh μZ x+ xmð Þð Þ

# $−xm ≤ x ≤ 0ð Þ

!vLμ2Z

sinh μZxmð Þcosh μZ 1+ xmð Þð Þ

sinh μZ 1−xð Þð Þ 0≤ x ≤ 1ð Þ

8>>><

>>>:, ð68Þ

396 LANDER ET AL.

4.3 | The LRNO approximation ofrobustness index

As the presence of a fixed non-receptor concentrationworks both for and against robust development, whetheror not the net effect is positive can only be seen from (theLRNO approximation ρZ of) our adopted measure ofrobustness !Rb. From Equation (46), we have

ρZ =1

B1 0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0eBZ xð Þ−B1 xð Þ½ (2dx

s

ð74Þ

or

f Z ρZð Þ=0, ð75Þ

where fZ(*) is fc(*) as defined in Equation (47) but withμc and ec replaced by μZ and

eZ =e

1+ h1Z=h0

sinh μZxmð Þcosh μ0 1+ xmð Þð Þsinh μ0xmð Þcosh μZ 1+ xmð Þð Þ

, ð76Þ

respectively.Note that unlike its counterpart Equation (46), the

right hand side of Equation (74) is independent of ρZ sothat it can be evaluated to get ρZ (without having to solvea nonlinear equation as we had to do for ρc). Upon writ-ing Equation (74) as

sinh2 μ0ð Þρ2Z =ð1

0I x;Zð Þ½ (2dx$G Zð Þ ð77Þ

where

I x;Zð Þ= eZsinh μZ 1−xð Þð Þ−sinh μ0 1−xð Þð Þ,

it is straightforward to evaluate the integral in Equa-tion (77) to obtain

2G Zð Þ= e2Zsinh 2μZð Þ

2μZ−1

) *+

sinh 2μ0ð Þ2μ0

−1) *

−2eZsinh μZ + μ0ð Þ

μZ + μ0−sinh μZ−μ0ð Þ

μZ−μ0

# $,

ð78Þ

Plotting G(Z) shows that ρ2Z is a convex function ofZ with a positive minimum at some minimum pointZmin> 0. Note that the convexity of ρZ can also be seenanalytically from the following properties of I(x; Z) forany x in [0, 1):

i. I x;0ð Þ= e−1ð Þsinh μ0 1−xð Þð Þ>0 0≤ x<1ð Þii. lim

Z!∞I x;Zð Þ½ (= −sinh μ0 1−xð Þð Þ<0 0≤ x<1ð Þ:

and, with μ2Z≥μ20 % 1,

iii. I x;Zð Þ monotone decreasing with increasing Z:

As a consequence, we have the following negativeresult similar to the corresponding finding in Refer-ence 25:

Proposition 8 The robustness of a signaling gradient in asteady state of LRNO deteriorates with increasing Zfor Z > Zmin.

For the illustrating example of Table 1, the optimalratio Zmin that gives the smallest ρZ is for Zmin ffi 1.092with ρmin ffi 0.0670 which is insignificantly belowρZ = 0.0693… for Z = 1. However, both are significantlybelow ρZ = 0.1365… for Z = 2 (even if the latter is stillbelow robustness threshold). Beyond an optimal level ofZ, continuing increase in non-receptor synthesis ratewould reduce the aberrant gradient concentration belowthe wild-type gradient !b1 xð Þ and worsen thecorresponding shape difference, eventually to an unac-ceptable level of robustness.

Superficially, this suggests that there is no need toconsider Z values higher than Zmin for a given problem,with Zmin ’ 1 giving rise to an acceptable level of gradi-ent shape distortion (as measured by our signal robust-ness index !Rb ). However, Zmin may have to be largerfor another problem for which there is a need for alarger amplitude reduction factor 1+ h1Z/h0. For exam-ple, Z would need to be 3 or larger for the same problemwith the larger ligand synthesis amplification factor(aberrancy) e = 4 (see Equation (72) or (73)). This sug-gests that the induced non-receptor synthesis should be

FIGURE 2 Effects of non-receptors

LANDER ET AL. 397

an increasing function Z(e) of the excess factor e. A largerZ value would induce a much more severe (and probablyunacceptable) signaling gradient shape distortion as seenfrom Equations (70) and (69). These observationsstrongly suggest the following conclusion consistent withthe simulation results of Reference 25:

Conclusion 9 A non-receptor synthesis rate in the formof Equations (51) and (55) does not offer a biologi-cally realistic mechanism for down-regulatingaberrant signaling except for moderate aberrancyso that Zmin ’ 1. In the latter case, robustness issensitive to any further increase in non-receptorconcentration.

Accurate numerical results for the exact solution(without the LRNO approximation) !bZ xð Þ and !Rb havebeen obtained for different (uniform) non-receptor syn-thesis rates (as characterized by the parameter Z) withadditional parameters associated with the non-receptorsassigned the following values in the illustrating example(also examined in Reference 45):

jdeg = kdeg, jon = kon, joff = 10koff , jN =10kR

It is evident from the results reported in Table 2 that ρZand eBZ 0ð Þ=Be 0;Zð Þ’ !be 0;Zð Þ are quite accurate approx-imations of the corresponding numerical solutions for!Rb Zð Þ and !bZ 0ð Þ= !be 0;Zð Þ of the new model for Equa-tion (51). As such, the effects of non-receptors are prettymuch delineated by the LRNO solution.

From either the computed LRNO solution or thenumerical solutions for the original nonlinear BVP, wesee that a moderate presence of non-receptors would gen-erally bring !bZ xð Þ= !be x;c,Zð Þ

' (c=0 closer to !b1 xð Þ but

generally renders !bZ xð Þ steeper and more convex than!b1 xð Þ as shown in Figure 2. (Note that the notation!be x;0,Z,1,0ð Þ in Figure 2 corresponds to !bZ xð Þ= !be x;Zð Þ.)The observations and conclusions above strongly suggestthat the inhibiting mechanism Equation (51) is not arealistic robustness promoting mechanism. It is examinedhere not only to elucidate the findings by numerical sim-ulations in Reference 25 but also, with the help of theexplicit analytical solution, to eliminate it as a strategy

for promoting signaling gradient robustness. This tenta-tive conclusion will be further strengthened by the devel-opments in the next two subsections.

4.4 | Addition of a negative feedback onreceptor synthesis

The LRNO solution (70) shows that the distortion of theslope and convexity of aberrant signaling gradient bynon-receptors is in the opposite direction relative to thatinduced by the feedback on signaling receptor synthesisrate given

μ2Z = μ20 +Zμ21 > μ20 >μ20

1+ cρmc= μ2c :

It would seem that we may be able to take advantage ofthis observation through a two-inhibitor model withreceptor and non-receptor synthesis rates of the form(27) and (51), respectively, to more effectively promoterobust gradients. The LRNO solution for such a model,denoted by !ae x;c,Zð Þ$ !aRZ xð Þ’ eARZ xð Þ, is straightfor-ward with

!be x;c,Zð Þ$ !bRZ xð Þ) eBRZ xð Þ= e!κ2 ρRZð Þα0

ARZ xð Þ

=e!νL

α0μ2RZ 1+ cρmRZð Þsinh μRZxmð Þ

cosh μRZ 1+ xmð Þð Þsinh μRZ 1−xð Þð Þ,

ð79Þ

for 0 ≤ x ≤ 1, where ρRZ is the LRNO approximation forthe robustness index !Rb for the present model, and

μ2RZ =μ20

1+ cρmRZ+Zμ21 ’

h01+ cρmRZ

+Zh1 ð80Þ

It is evident from the expression for μ2RZ that itsnumerical value for a given gradient system is not sub-stantially different from μ2Z ’ h0 +Zh1 for c = O(1) sinceρRZ< ρZ = O(10−1) for Z = O(1). This observation is con-firmed by accurate numerical solutions for the newmodel (with and without the LRNO approximation)

TABLE 2 (g1 = 0.2, h1 = 10, f1 = 0.01, gn = 10, vL = 0.05, e = 2)

Z !Rb ρZ !b2 0,Zð Þ eB2(0; Z) !b1 0,0ð Þ B1(0, 0)

0 0.3943 0.3938 0.11517 0.1169 0.05790 0.0584

1 0.0714 0.0693 0.07398 0.0739 0.05790 0.0584

2 0.1298 0.1365 0.05597 0.0555 0.05790 0.0584

398 LANDER ET AL.

reported in Table 3 for the same sample problem as inTables 1 and 2. The results show that there is not anappreciable reduction in the robustness index with c>0for Z = 1. The minimal effect of the signaling gradientitself is illustrated in Figure 3 for !be x;c,Zð Þ. These obser-vations are recorded as:

Conclusion 10 In the presence of non-receptors (as aninhibiting agent for an aberrant signaling gradient)in the form of the synthesis rate (51) and (55), the(steady state limit of a) spatially uniform non-localnegative feedback (27) on receptor synthesis ratereduces the signaling gradient concentration andameliorates the distortion of signaling gradientshape but not appreciably and only for Z , 1. ForZ - 2, the robustness index !Rb ’ ρZ actually deteri-orates with the addition of the negative feedback onreceptor synthesis rate.

Similar to the case without the negative feedbackon receptor synthesis, the results in Table 3 alsoshow that too high a non-receptor-to-receptor synthe-sis rate ratio (Z ≥ 2 in our example) would cause anexcessive reduction of signal morphogen concentra-tion and too severe a shape distortion to result inan unacceptable robustness index. For Z ≥ 2 (neededfor e > 2), the addition of feedback on receptor syn-thesis rate is actually deleterious to robust develop-ment for the illustrating example. The slightreduction of the shape distortion parameter μ2RZ for apositive c, does not compensate for the considerablylarger reduction of the amplitude reduction factor (thatdrives the amplitude of the signaling gradient at x = 0 towell below the concentration of the wild-type gradient atthe same location). The corresponding graphs of!bRZ xð Þ= !be x;c,Zð Þ for different Z and c in Figure 3 clearlyshow why the robustness index !Rb may eventually deteri-orate with increasing Z.

To confirm, we have from

ρRZ =1

B1 0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0eBRZ xð Þ−B1 xð Þ½ (2dx

s

ð81Þ

the equation for determining ρRZ,

f RZ ρRZð Þ=0, ð82Þ

where fRZ(*) is fc(*) as defined in Equation (47) but withμc and ec replaced by μRZ (with m taken to be 1 in Equa-tion (27) as before) and

eRZ =e

1+ 1+ cρRZð Þh1Z=h0sinh μRZxmð Þcosh μ0 1+ xmð Þð Þsinh μ0xmð Þcosh μRZ 1+ xmð Þð Þ

,

ð83Þ

respectively. Solutions for some typical ρRZ calculatedfrom Equation (82) are shown in Table 3 to confirm theobservations made above and re-affirm Conclusion 9.

TABLE 3 (g0 = g1 = 0.2, h0 = h1 = 10, gr = 1, gn = 10; f0 = 0.001, f1 = 0.01, vL = 0.05, e = 2)

Z c !Rb ρRZ !bRZ 0ð Þ BRZ(0) !b1 0ð Þ

0 0 0.3943 0.3938 0.11517 0.1169 0.0579

0 1 0.3557 0.3565 0.10222 0.1062 0.0579

0 2 0.3269 0.3284 0.09407 0.0959 0.0579

1 0 0.0714 0.0693 0.07398 0.0739 0.0579

1 1 0.0619 0.0612 0.07550 0.0710 0.0579

1 2 0.0560 0.0566 0.06894 0.0688 0.0579

2 0 0.1298 0.1365 0.05597 0.0555 0.0579

2 1 0.1471 0.1547 0.05778 0.0497 0.0579

2 2 0.1683 0.1766 0.04463 0.0449 0.0579

FIGURE 3 Effects of negative feedback on receptor synthesisfor Z > 0

LANDER ET AL. 399

4.5 | Root-mean-square displacementdifferential

With μRZ and eRZ both dependent on ρRZ, the relation(82) is now a highly nonlinear equation for ρRZ. Still, theexpression for the new shape distortion factor μRZ inEquation (80) shows clearly that it is not possible for thenegative feedback on receptor synthesis rate to reducethe shape distortion to an acceptable level if the non-receptor to receptor ratio Z should be much higher than1. That this is not reflected in the computed values ofρRZ in Table 3 suggests that the signaling robustnessindex Rb is by itself not always an adequate measure ofrobustness. For this reason, we have also introduced inReferences 16 and 25 its companion robust index Rx

(denoted by L in Reference 25) that measures the root-mean-square displacement differential of the aberrantsignaling gradient.

Let xe(b) and x1(b) be the location where the aberrantand wild-type gradients attains the value b, respectively,that is, !bc xeð Þ= !b1 x1ð Þ= b . In steady state, the displace-ment robustness index !Rx is defined by

!Rx =1

xℓ−xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

bh−bℓ

ðbh

bℓxe bð Þ−x1 bð Þ½ (2db

s

: ð84Þ

To minimize the effects of outliers, we maylimit the range of b to be the interval (bℓ, bh) with0≤ bℓ < bh ≤ !b 0ð Þ . (We may take bℓ = !b 0ð Þ=10 andbh =9!b 0ð Þ=10 for instance.)

For gradients in a steady state of LRNO, the depen-dence of displacement Δx = xe(b) − x1(b) on any feedbackfor a non-negative range of xe and x1 is through theexpression (see Equation (70))

xe =1−1μx

sinh−1 bβx

) *, x1 = 1−

1μ0

sinh−1 bβ0

) *

where μ2x is μ2RZ with ρRZ replaced by ρx (the LRNOapproximation for !Rx) and

βx =e!vLκ2

α0μ2x

sinh μxxmð Þcosh μx 1+ xmð Þð Þ

, β0 =!vLα0μ20

sinh μ0xmð Þcosh μ0 1+ xmð Þð Þ

:

The LRNO approximation of !Rx is then

!Rx ’ ρx =1

xℓ−xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

bh−bℓ

ðbh

bℓ

1μ0

sinh−1 bβ0

) *−

1μx

sinh−1 bβx

) *- .2db

s

:

ð85Þ

With μ20 % 1, we may, to a good first approximation,work with the asymptotic values of these expressions to get

Δx= c0 + c1ln bð Þ ð86Þ

where

c0 =1μRZ

lnβRZ2

) *−

1μ0

lnβ02

) *, c1 =

1μ0

−1μRZ

) *: ð87Þ

It follows that

For sample calculations, we take bh = b(0) and bℓ = b(1) ≈ 0 so that

ρ0 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic20 + 2c0c1 ln bhð Þ−1f g+ c21 ln bhð Þf g2−2 ln bhð Þ−1f g

+ ,q

ð88Þ

For more moderate values of h0 = h1 (withμ0 =

ffiffiffiffiffih0

p=O 3ð Þ), we would work with the exact inverse

functions

xe bð Þ=1−1μx

ln qx bð Þð Þ, x1 bð Þ=1−1μ0

ln q0 bð Þð Þ ð89Þ

in Equation (85) where

qx bð Þ= bβx

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibβx

) *2

+ 1

s

!Rx ’ ρx ’ ρ0 =1

xℓ−xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

bh−bℓ

ðbh

bℓc0 + c1ln bð Þ½ (2db

s

=1

xℓ−xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

bh−bℓc20b+2c0c1b ln bð Þ−1f g+ c21 b ln bð Þf g2−2ðb ln bð Þ−1f g

+ ,' (bhbℓ

r

400 LANDER ET AL.

and q0(b) is qx(b) with βx replaced by β0. Table 4 reportssome typical results for the illustrating example ofTables 1–3 by the exact inverses (89):

With robustness measured by the new robustnessindex, the results for the illustrative example computedfrom the LRNO solution in Table 4 now show clearly howρx deteriorates with increasing c even for typical values ofZ around Zmin (while ρRZ deteriorates with increasingc only for Z > Zmin). They further strengthen Conclusionand complement it with the following (instead of Conclu-sion 10):

Conclusion 11 In the presence of the non-receptorinhibiting agent of the form (51) and (55), the(steady state limit of a) negative spatially uniformnon-local feedback (27) on receptor synthesis ratemore readily exacerbates the aberrancy of the sig-naling gradient concentration when measured bythe displacement robustness index ρx.

Given the different (and sometimes opposite) wayshow the negative feedback on receptor synthesis rateimpacts the two robustness indices of an aberrant gradientfor Z ≥ 0, we should generally measure robustness by cal-culating both robustness indices before reaching any con-clusion about the robustness of the signaling gradient.

5 | FEEDBACK ON NON-RECEPTOR SYNTHESIS RATES

5.1 | The model

Non-receptors as an inhibiting agent for down-regulatingan aberrant gradient in the form (51) led to Conclusions-11 because the required synthesis rate has the conse-quence of increasing the shape distortion parameterseverely for Z > 1 (which is needed for aberrancy factore > 2). This would not be the case if the level of feedbackdepends on deviation from the wild-type gradient as mea-sured by (one of) the robustness indices. For this purpose,we consider a positive feedback on non-receptor synthe-sis rate V N that also depends on the signaling robustnessindex of the form

VN = !VN η+ σRsb t−τð Þ

' (H t− teð Þ ð90Þ

where the parameter σ and s characterize the strengthand sensitivity of the feedback and the parameter τ per-tains to a possible time delay on the effect of feedback(and te > 0 is again the instant of onset of the genetic orepigenetic perturbations). The model in the previoussection corresponds to η = 1 and σ = 0 while the caseη = 0 and σ > 0 offers a positive feedback for stimulatinga non-receptor synthesis rate commensurate with theaberrancy measured by the robustness index Rb. The sub-sequent development of the theory for this new feedbackprocess is similar if the signaling robustness index Rb isreplaced by the corresponding displacement robustnessindex Rx as defined in Equation (84).

Anticipating a limiting time-independent steady stateof signaling gradient, we expect

VN ! !VN η+ σ!Rsb

' (= !VN!κ2N η,σð Þ s>0ð Þ ð91Þ

as the system approaches a time independent steadystate. To maximize the effect of !Rb in the feedback (91),the parameter s will be set to 1 thereafter (similar to set-ting m = 1 in the feedback on receptor synthesis rate(27)). For that reason, the parameter s will not be dis-played explicitly as a parameter of !κ2N in Equation (91)and elsewhere. Given Conclusion 9 and the fact that aninstantaneously induced fixed synthesis rate is ratherunrealistic, we limit our discussion to the η = 0 case. Anaberrancy dependent positive feedback on non-receptorsynthesis is also more consistent with experimentalobservations reported in references cited in the lastsection.

Upon modifying the model of Sec. 3 with the feed-back on the receptor synthesis rate (27) to include thefeedback (90), we reduce the new steady state problem tothe following BVP for !aRN xð Þ= !ae x;c,Z,η,σð Þ:

!a00RN −!κ2g0!aRN

α0 + ζ0!aRN−Z

!κ2Ng1!aRNα1 + ζ1!aRN

+ e!vLH −xð Þ=0, ð92Þ

!a0RN −xmð Þ=0, !aRN 1ð Þ=0: ð93Þ

TABLE 4 (g0 = g1 = 0.2, h0 = h1 = 10, gr = 1, gn = 10, f0 = 0.001, f1 = 0.01, vL = 0.05, e = 2)

Z 0 0 0 1 1 1 2 2 2

c 0 1 2 0 1 2 0 1 2

ρRZ 0.3938 0.3565 0.3284 0.0693 0.0612 0.0566 0.1365 0.1547 0.1766

ρx 0.1987 0.2095 0.2159 0.0576 0.0575 0.0581 0.1454 0.1579 0.1737

LANDER ET AL. 401

Correspondingly, the remaining gradients are given by

!bRN xð Þ=!κ2!aRN

α0 + ζ0!aRN, !rRN xð Þ=

!κ2α0α0 + ζ0!aRN

ð94Þ

!cRN xð Þ=!κ2N!aRN

α1 + ζ1!aRN, !nRN xð Þ=

!κ2Nα1α1 + ζ1!aRN

ð95Þ

with !bRN xð Þ= !be x;c,Z,η,σð Þ, etc. For a particular wingimaginal disc, it is clear from the model that therobustness of its development depends on the strengthof the feedback on receptor synthesis rate character-ized by the parameter c and the strength of the non-receptor synthesis rate characterized by the parame-ters Z, η and σ.

5.2 | Low receptor and non-receptoroccupancy

When both receptors and non-receptors are in a state oflow occupancy, we may linearize the ODE (92) to get for!aRN xð Þ’ eARN xð Þ

A00RN −μ2RNARN +!vLH −xð Þ=0, ð96Þ

subject to the two end conditions (93) with

μ2RN = μ2R +Zμ2N = !κ2 ρRNð Þμ20 +Z!κ2N ρRNð Þμ21

’ h01+ cρRN

+Zh1 η+ σρRNð Þ,ð97Þ

where ρRN is the LRNO approximation of !Rb for the pre-sent model and we have made the highly accurateapproximations μ20 = h0 and μ21 = h1 consistent with thesimplification made in the earlier sections and in Refer-ence 22.

The exact solution for ARN(x) ≡ Ae(x; c, Z, η, σ) isgiven by

In the range 0 ≤ x ≤ 1, we have

!bRN xð Þ’ eBRN xð Þ= e!κ2 ρRNð Þα0

ARN xð Þ

=e!κ2 ρRNð Þ!vLα0μ2RN

sinh μRNxmð Þcosh μRN 1+ xmð Þð Þ

sinh μRN 1−xð Þð Þð99Þ

where μ2RN is as given in Equation (97). From theexpression

α0!bRN 0ð Þ’ e!κ2 ρRNð ÞARN 0ð Þ

=e!vLμ2RN

!κ2 ρRNð Þsinh μRNxmð Þcosh μRN 1+ xmð Þð ÞÞ

sinh μRNð Þ:

we obtain

!bRN 0ð Þ ’ eBRN 0ð Þ

’ e!vL=g0 1−e−2μRNxmð Þ1+Z 1+ cρRNð Þ η+ σρRNð Þ h1=h0ð Þ

ð100Þ

given fk & gk for both k = 0 and 1. Correspondingly,we have

!bRN 0ð Þ!b1 0ð Þ

’ eBRN 0ð ÞB1 0ð Þ

~e 1−e−2μRNxmð Þ= 1−e−2μ0xmð Þ1+Z 1+ cρRNð Þ η+ σρRNð Þ h1=h0ð Þ

O 1ð Þ

ð101Þ

It is evident from Equations (100) and (101) that, inthe presence of non-receptors, a spatially uniform non-local positive feedback of the type (90) with η > 0 orσ > 0 (with Z = 1) reduces the magnitude of the signalinggradient (toward the wild type gradient) but also inducesa shape change relative to that without any feedback.While this effect on the gradient shape counters that ofthe negative feedback on receptor synthesis, it may over-compensate the latter (as delineated in the previous sec-tion) if the magnitude of η should be too large relative to

ARN xð Þ=

!vLμ2RN

1−cosh μRNð Þ

cosh μRN 1+ xmð Þð Þcosh μRN x+ xmð Þð Þ

# $−xm ≤ x ≤ 0ð Þ

!vLμ2RN

sinh μRNxmð Þcosh μRN 1+ xmð Þð Þ

sinh μRN 1−xð Þð Þ 0≤ x ≤ 1ð Þ

8>>><

>>>:ð98Þ

402 LANDER ET AL.

what is needed to restore the signaling gradient to thecorresponding wild-type gradient. However with η = 0,the non-receptor synthesis rate needed to achieve robust-ness would be adjusted by the level of aberrancy of thesignaling gradient through the feedback and the resultingdown-regulation of the aberrant gradient would be self-limiting as shown by exact numerical solutions for signal-ing gradient in Figure 4. Hence the feedback process ismore likely to be successful in keeping the aberrant sig-naling gradient close to the wild-type gradient prior tothe exogenous perturbations.

The net results of the different effects due to a positivefeedback on non-receptor synthesis rate with η = 0 andσ > 0 is reflected in the LRNO robustness index

ρRN =1

B1 0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1

0eBRN xð Þ−B1 xð Þ½ (2dx

s

ð102Þ

or

f RN ρRNð Þ=0, ð103Þ

where fRN(*) is fc(*) as defined in Equation (47) but withμc and ec replaced by μRN (with m and s both taken tobe 1) and

eRN =

e1+ 1+ cρRNð Þ η+ σρRNð Þh1Z=h0

sinh μRNxmð Þcosh μ0 1 + xmð Þð Þsinh μ0xmð Þcosh μRN 1+ xmð Þð Þ

,

ð104Þ

respectively. With μRN and eRN both depending on ρRN,the relation (103) is now a highly nonlinear equationfor ρRN.

5.3 | Robustness dependent positivefeedback on non-receptor synthesis

With the impact of a robustness index dependent positivefeedback on non-receptor synthesis being self-limiting,we examine in more detail first the consequences of theLRNO results for such a feedback alone. These resultsare obtained from those of the previous section by settingη = 0 and c = 0. The signaling morphogen concentration!bRN 0ð Þ' (

c= η=0, denoted by !bN xð Þ’ eBN xð Þ, is given by

!bN xð Þ’ eBN xð Þ= eα0

AN xð Þ x≥0ð Þ

=e!vL=g0

1+ZσρNh1=h0

sinh μNxmð Þcosh μN 1+ xmð Þð Þ

sinh μN 1−xð Þð Þ

ð105Þ

where ρN = [ρRN]c=η=0 is the LRNO approximation ofthe robustness index and

μ2N = μ20 +Zμ2N = h0 +Zh1σρN : ð106Þ

The signaling ligand concentration at x = 0 is

!bN 0ð Þ’ eBN 0ð Þ= e!νLg0

λN1+ZσρNh1=h0

where

λN =sinh μNxmð Þsinh μNð Þcosh μN 1+ xmð Þð Þ

:

For the example used throughout this article, we haveh1/h0 = 1 and therewith

μ2N = h0 1+ZσρNð Þ: ð107Þ

With h0 % 1 (but μ0 = O(1) as in our illustrative example,we have from Equations (105) and (107))

!bN 0ð Þ!b1 0ð Þ

’ eBN 0ð ÞB1 0ð Þ

’ e 1−e−2μNxmð Þ1+ZσρNð Þ 1−e−2μ0xmð Þ

: ð108Þ

The LRNO approximation of the relative magnitude!bN 0ð Þ=!b1 0ð Þ depends on the robustness index through theshape distortion parameter μN and the amplitude reduc-tion factor 1+ZσρNh1/h0. The LRNO approximation ρNof the (signaling) robustness index is determined by

f N ρNð Þ=0, ð109Þ

where fN(*) is fc(*) as defined in Equation (47) but with μcand ec replaced by μN (with m and s both taken tobe 1) and

FIGURE 4 Positive feedback on non-receptor synthesisonly (c = η = 0)

LANDER ET AL. 403

eN =e

1+ZσρN

sinh μNxmð Þcosh μ0 1+ xmð Þð Þsinh μ0xmð Þcosh μN 1+ xmð Þð Þ

’ e1+ZσρN

1−e−2μNxm

1−e−2μ0xmμN≥μ0 % 1ð Þ,

ð110Þ

respectively (having specialized to the case h1 = h0).The following observations are immediate from

Equation (108):

• The amplitude reduction factor 1 + ZσρN is consider-ably smaller than the corresponding factor for themodel with the non-receptor synthesis rate (51) whenZσ is O(Z) for the latter model (or O(Zη) in the feedback(91)) since ρN is expected to be considerably less thanunity.

• The shape distortion parameter μN is considerablysmaller than the corresponding parameter μZ so thatthe shape of !bN xð Þ’ eBN xð Þ is less steep and less con-vex than !bZ xð Þ’ eBZ xð Þ.

By the first observation above, the reduction of the sig-naling differential robustness index !Rb ’ ρN for Zη = 0(and Zσ = 1) is expected to be considerably more modestthan the corresponding reduction of !Rb ’ ρZ for Zη = 1(and Zσ = 0). In particular, it barely meets the conserva-tive threshold of ρN≤ 0.2 with ρN = 0.1972 for Zσ = 2 forour illustrative example. This is understandable given theamplitude reduction factor is now (1+ZσρRN)

−1 (forη = 0) instead of (1+Z)−1 (for η = 1 and σ = 0) with e/(1+Z) = 1 for e = 2 and Z = 1. We need σρN = O(1),i.e., σ = O(5), for the reduction to be comparable to theη = 1 (and σ = 0) case. On the other hand, the shape dis-tortion (for η = 0 and σ = 1) is now less severe withμ2N = h0 1+ZσρNh1=h0ð Þ< h0 1+Zh1=h0ð Þ since ρN<1 forsome degree of robustness. The actual benefit (or cost)for a η = 0 and σ = 1 feedback on non-receptor synthesisrest on the net effect from the two opposite bulletedimpacts above on the two robustness indices.

While this net effect may be case specific, it shouldbe evident from the expression (107) that the impact ofany increase in the synthesis rate ratio Zσ is much lessthan full (as it would be for the synthesis rate (51)) asonly a fraction ρN of Zσ is felt by the gradient system.Moreover, that fraction would be further reducedby the impact of the increase in Zσ on reducing therobustness index. In other words, the impact of anyincrease in Zσ is doubly palliated (as long as the robust-ness index is less than unity), thereby reducing its effecton the amplitude reduction factor and the shape distor-tion parameter μN. With the signaling gradient unlikelyto overshoot that of the target wild-type gradient (in con-trast to Figure 3), the impact of non-receptors as an

inhibiting agent introduced through the feedback (91) isseen to be self-limiting, at least more so than through (51).This enables us to assert the following conclusion:

Conclusion 12 The feedback (91) with η = 0 is expectedto be self-limiting in correcting aberrancy induced bygenetic or epi-genetic perturbations and consequentlybiologically more realistic than η > 0 (and σ = 0) fordown-regulating the aberrant signaling gradient.

Remark 13 For σ > 0, the addition of η > 0 wouldfurther decrease the amplitude reduction factor to(1 + Z(η + σρRN))−1. However, the gain, when notdeleterious, is offset by a more severe shape distor-tion resulting from μ2RN = h0 1+Z η+ σρRNð Þf g>h0 1+ZσρRNf g.

5.4 | An effective multi-feedbackinstrument for robust development

With μZ > μN ≥ μ0 > 1 ≥ μc, the shape distortion inducedby the feedback (91) with η = 0 is opposite to that by thenegative feedback (33) on the receptor synthesis rate. Act-ing alone, the latter feedback has no impact on the ampli-tude reduction factor. Administering concurrently withthe positive feedback on non-receptor, the two mitigatingeffects on shape distortion should help to lower the robust-ness index of the aberrant signaling gradient due to eachfeedback acting alone. To take advantage of thisobservation, we specialize the general results of Sub-section 5.2 to the case η = 0 so that the positive feedbackon non-receptor is robustness dependent. The signalingmorphogen concentration !bRN 0ð Þ

' (η=0 , to be denoted by

!bRS xð Þ’ eBRS xð Þ, is given by

!bRS xð Þ’ eBRS xð Þ= e!κ2 ρRSð Þα0

ARS xð Þ x≥0ð Þ

=e!vL=g0

1+ZσρRS 1+ cρRSð Þh1=h0sinh μRSxmð Þ

cosh μRS 1+ xmð Þð Þsinh μRS 1−xð Þð Þ

ð111Þ

where ρRS = [ρRN]η = 0 is the LRNO approximation of therobustness index and

μ2RS = μ2c +Zμ2S =h0

1+ cρRS+Zh1σρRS ð112Þ

Accurate numerical solutions for the signalinggradient have been obtained for the illustrative exam-ple used throughout this paper and shown in Figure 5for several combinations of feedback parameter values.Comparison with the corresponding graphs in Figure 4shows the benefit of the multi-feedback approach.

404 LANDER ET AL.

5.4.1 | The LRNO signaling morphogenconcentration eBRS(0)

How the multiple feedback process further improvesthe down-regulation of the aberrancy induced by thegenetic or epigenetic perturbations can be glimpsed byexamining the LRNO approximation of the exact solu-tion for the new model. Without going through stepssimilar to the previous models, we simply report herethat the LRNO signaling morphogen concentration atx = 0 is given by

eBRS 0ð Þ= e!νL=g01+ZσρRS 1+ cρRSð Þh1=h0

λRS ð113Þ

where

λRS =sinh μRSxmð Þsinh μRSð Þcosh μRS 1+ xmð Þð Þ

: ð114Þ

For the example used throughout this article, we haveh1/h0 = 1 and therewith

μ2RN' (

η=0 = h01

1+ cρRS+ZσρRS

h1h0

) *$ μ2RS ð115Þ

and

!bRS 0ð Þ!b1 0ð Þ

) e 1−e−2μRSxmð Þ1+ZσρRS 1+ cρRSð Þh1=h0ð Þ 1−e−2μ0xmð Þ

: ð116Þ

It is evident from Equation (116) that the amplitudereduction factor is now larger than that for c = 0 so thatit has the effect of reducing !bN 0ð Þ< !be 0ð Þ . At the same

time, we have μ2RS < μ2N which reduces the shape distor-tion caused by Zσ>0. Together, they enable us toconclude:

Conclusion 14 Concurrent applications of the positivefeedback on non-receptor synthesis (91) with η = 0and a (moderate strength) negative feedback onreceptor synthesis rate !κ2 !Rbð Þ!vR promote moreeffectively robustness of a signaling gradient thaneither feedback acting alone.

For a sufficiently large value of c > 0, μ2RS may bereduced to nearly μ20 so that the resulting aberrant gradi-ent shape would approach that of the wild type. Withμ2RS ffi μ20 for c in the range that renders

1+ cρRSð Þ 1−ZσρRSh1h0

) *ffi 1, ð117Þ

we have

eBRS 0ð Þ’ e!νLg0

1−ZσρRSh1h0

) *sinh μ0xmð Þsinh μ0ð Þcosh μ0 1+ xmð Þð Þ

=e!νLλRS

g0 1+ cρRSð Þsinh μ0xmð Þsinh μ0ð Þcosh μ0 1+ xmð Þð Þ

:

ð118Þ

Whether the resulting aberrant signaling gradient is suffi-ciently close to the wild type can only be read from thecorresponding two robustness indices.

5.4.2 | The LRNO robustness index

The LRNO approximation of the corresponding differen-tial signaling robustness index ρRS is determined by

f RS ρRSð Þ=0, ð119Þ

where fRS(*) is fc(*) as defined in Equation (47) but with μcand ec replaced by μRS (with m and s both taken to be1) and

eRS =e

1+ZσρRS 1+ cρRSð Þsinh μRSxmð Þcosh μ0 1+ xmð Þð Þsinh μ0xmð Þcosh μRS 1+ xmð Þð Þ

ð120Þ

’ e1+ZσρRS 1+ cρRSð Þ

1−e−2μRSxm

1−e−2μ0xm, ð121Þ

respectively (having specialized to the case h1 = h0). Therelations (119) and (120) also apply to the corresponding

FIGURE 5 Feedback on receptor and non-receptor synthesis(c > 0, η = 0)

LANDER ET AL. 405

differential displacement robustness index ρx if we replaceρRS in !κ2, !κ2N , μ

2RS, eRS and fRS by ρx.

One effect from the addition of a negative feedbackon signaling receptor synthesis rate is to increase theamplitude reduction factor to 1 + ZσρRS(1 + cρRS)h1/h0and thereby decreases the “amplitude” of eBRS(x) (seeEquation (113)). Generally, this effect should reduce ρRS.However, it is not difficult to see that.

Lemma 15 ρRS does not tend to 0 as c ! ∞.

Proof Assuming the opposite so that ρRS ! 0 as c ! ∞,there are three possibilities pertaining to the magni-tude of cρRS:

1. cρRS! 0 with

eBRS xð Þ! eB1 xð Þ:

It follows that ρRS ! ρ0 > 0 as given in Equation (26)contradicting the assertion to the contrary.

2. There exists a positive ϖ such that ρRS ! ϖ > 0 and

eBRS xð Þ= e!vLg0

sinh μRSxmð Þcosh μRS 1+ xmð Þð Þ

sinh μRS 1−xð Þð Þ

with

μ2RS =μ20

1+ϖ< μ20:

It follows that eBRS(x) is generally not B1(x) andρRS > 0 as c ! ∞, contradicting the assertion to thecontrary.

3. cρRS ! ∞ but cρ2RS !ϖ , then we have 1+ZσρRS(1+ cρRS)h1/h0! 1+Zσϖh1/h>1 and μ2RS ! 0< μ20 sothat ρRS must again be positive in the limit as c!∞and we again arrive at a contradiction.

As a consequence of the lemma above, we have thefollowing observations on the robustness index:

• For a fixed value of Zσ (and η = 0), ρRS must eventuallyreach a minimum and begin to increase with furtherincrease in c tending to a finite limit ρ∞. Since the limitρ∞ depends on the value of Zσ, the specific two-feedback mechanism is said to be self-limiting.

• For any moderate value of Zσ (and η = 0) to result in acorresponding ρN, the following inequalities on theshape distortion parameter and the amplitude reduc-tion factor hold

μ201

1+ cρN+ZσρN

) *< μ20 1+ZσρNð Þ= μ2N ,

1 +ZσρN 1+ cρNð Þh1h0

> 1+ZσρNh1=h0 :

Hence, the addition of a moderate strength negativefeedback on receptor synthesis rate should bring theaberrant signaling gradient closer to the wild typewith ρRS < ρN.

• For a fixed moderate value of Zσ so that ZσρN < 1, asufficiently large value of c, say c1, would render

11+ cρN

+ZσρN ≈1,

and keep the gradient shape close to the wild-typegradient shape. Increasing c well beyond c1 would

TABLE 5 (g0 = g1 = 0.2, h0 = h1 = 10, gr = 1, gn = 10 f0 = 0.001, f1 = 0.01, vL = 0.05, e = 2, η = 0)

c\Zσ 0 1 2 4

0 0.3938 0.2539 0.1972 0.1421 ρRZ

0.1987 0.1514 0.1249 0.0949 ρx

1 0.3565 0.2263 0.1769 0.1289 ρRZ

0.2095 0.1526 0.1239 0.0930 ρx

2 0.3284 0.2073 0.1626 0.1194 ρRZ

0.2159 0.1521 0.1223 0.0911 ρx

4 0.2872 0.1815 0.1431 0.1059 ρRZ

0.2187 0.1483 0.1180 0.0872 ρx

406 LANDER ET AL.

reduce the shape parameter of the modified gradientwell below unity, thereby would distort the gradientunduly in the opposite direction and work againstrobustness.These results provide a specific realization of how an

appropriate two-feedback combination of receptor andnon-receptor synthesis rates may enhance both aberrantsignaling ligand concentration reduction and shape-change amelioration:

Conclusion 16 A multi-feedback instrument of the type(27) and (91) with η = 0 is both effective and self-lim-iting for down-regulating the aberrant signaling gra-dient without distorting unacceptably the slope andconvexity of the wild-type signaling gradient.

The development above applies also to the displace-ment differential robustness index ρx. As such the samequalitative conclusion may also be said about both indices.And both should be examined for robust signaling sincethey measure different features of the signaling gradient.Table 5 gives the two indices for the illustrative examplefor a range of c and Zσ showing the benefits of an appro-priate combination of the multi-feedback instrument.

Remark 17 Similar to the c = 0 case, the addition ofη > 0 to the present two-feedback system (of c > 0and σ > 0) would be a two edge sword: a furtherreduction of the amplitude reduction factor on theone hand, and an increase of the shape distortionfactor μ2RN on the other hand. The former does notalways promote robustness as it is not self-limitingand may down-regulate the signaling gradient to alevel substantially below the wild-type gradient.

6 | CONCLUDING REMARKS

When an abnormal genetic or epigenetic perturbationinterrupts an ongoing biological development, one ormore agents that counteract the unwelcome effects of theinduced signaling distortion need to be activated by theonset of abnormal development to down-regulate theaberrancy. This means the existence of some kind of feed-back process in order to promote robust signaling. Feed-back has long been seen as a mechanism for attainingrobust biological development and specific feedbackloops have been identified in the morphogen literaturesuch as Reference 13, 19, 23, 24, and 29 and others.Though the conventional Hill function type (theoretical)negative feedback on receptor synthesis rate proves to beineffective for this purpose,17,25,26 we have shown in

Reference 22 that a multi-feedback strategies involvingdirect (robustness index induced) reduction of morpho-gen synthesis rates are capable of promoting robust sig-naling. To the extent that feedback mechanisms may notdown-regulate morphogen synthesis rate directly, weneed to determine a more realistic multi-feedback mech-anism for robust signaling.

Among the possible mechanisms for achieving robustsignaling that are biologically meaningful and realistic,the potential of non-receptors down-regulating signaling(and hence promoting robust development) has alreadybeen established theoretically in References 17 and 18.While numerical simulations in Reference 25 show thatHill function type feedback involving non-receptors oftenresults in gradient systems that are either still unaccept-ably aberrant or biologically unrealistic, we show in thisarticle that a biologically realistic multi-feedback strategyinvolving a positive feedback on non-receptors andanother known feedback process (such as a negative feed-back on receptor synthesis rate) exists and is effective (ina self-limiting way) in promoting signaling gradientrobustness. The result also suggests that other combina-tions of known feedback processes should be explored.

Understanding how robust signaling can be attainedby multi-feedback mechanisms is important not only toshed light on the reliability of developing signaling gradi-ent systems, but also to help explain the ubiquitous pres-ence of the many elaborate regulatory schemes inmorphogen systems.

ACKNOWLEDGMENTSThe research of Q. Nie was partially supported by a NSFgrant DMS1763272 and a grant from the Simons Founda-tion (594598, QN) awarded to UCI. The research ofA.D. Lander was supported in part by NIH grantP50-GM-076516 awarded to UCI. The work ofF.Y.M. Wan was supported NSF (UBM) DMS-1129008awarded to UCI. The work of C. Sanchez-Tapia wassupported by an AFOSR grant FA9550-14-1-0060awarded to Rutgers University.

ORCIDFrederic Y. M. Wan https://orcid.org/0000-0002-5184-1227

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How to cite this article: Lander AD, Nie Q,Sanchez-Tapia C, Simonyan A, Wan FYM.Regulatory feedback on receptor and non-receptorsynthesis for robust signaling. DevelopmentalDynamics. 2020;249:383–409. https://doi.org/10.1002/dvdy.160

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